EXAMPLES. 251 



system can start from rest and move so that each cylinder rolls on the table 

 with the constant acceleration 



Mg sin 2a (1 + cos 2a)/{w 1 (1 + ^ 2 /V) + 2 ( l + ^W) + 4 ^ sin2 2a }> 

 where B,ma=(r 1 ~r 2 )/d, and d is the initial distance between the axes of the 

 cylinders. 



25. A circular cylinder of radius a whose centre of inertia is at a distance 

 b from its axis rolls on a horizontal plane. Prove that if it oscillates its 

 angular motion is given by an equation of the form 



+6 2 - 2ab cos 6}=gb (cos 6- cos a). 



26. A curve is drawn on the convex side of a hypocycloid parallel to it, 

 the distance between the curves being a, and a circular disc of radius a rolls 

 on the concave side under the action of a force to the centre varying as the 

 distance. Prove that the motion is isochronous for oscillations of all 

 amplitudes. 



27. On the top of a fixed smooth sphere rests a fine uniform ring with 

 its centre in the vertical diameter, and its diameter subtends an angle 2a at 

 the centre of the sphere. Prove that, if the ring is slightly displaced, it will 

 first begin to leave the sphere when its plane has turned through an angle Q 

 given by the equation 



sin (0 + a) sin a = 2 cos 2 a (2-3 cos 0). 



28. A uniform rod lying at rest in a smooth sphere is of such length that 

 it subtends a right angle at the centre. The rod is projected so that its ends 

 remain on the sphere and make complete revolutions in a vertical plane. 

 Prove that, if V is the initial velocity of the centre, and a the radius of the 

 sphere, 



29. Two uniform rods of equal length (a*/ 2 ) an ^ of equal mass are 

 firmly fixed at one extremity of each and are at right angles. The rods are 

 placed over two smooth pegs, distant c apart, in a horizontal line and move in 

 the vertical plane through the pegs. Prove that the angular motion of the 

 right angle on the semicircle it describes is given by one of the equations 



<p 2 (%a* ac cos^0-f c 2 ) + 4g(a cos-|< c cos <) = const., 



and if the motion is a small oscillation the length of the simple equivalent 

 pendulum is 



(2a 2 + 3c 2 ~ 3c) (4c ~ a). 



30. Two equal uniform rods of mass m and length 2a are free to turn 

 about their middle points which are fixed at a distance 2a apart in a horizontal 

 line. The rods being horizontal a uniform sphere of mass M and radius c is 

 gently placed upon them at the point where their ends meet. Prove that, 

 if 9J/"{a 2 +c 2 } 2 =2m{a 2 -c 2 } 2 , the sphere will, as it leaves the rods, have 

 half the velocity which it would have had after falling freely through the same 

 height. 



